Is the cosmic no hair conjecture true in the Einstein-Maxwell-Dilaton system?

We investigate the gravitational collapsing phenomenon in the spherically s ymmetric Einstein-Maxwell-dilaton system with a positive cosmological const ant. As a preparation, we first study some general properties of the horizo ns in asymptotically de Sitter spacetime and prove that the area of the: ho rizons does not decrease and has an upper bound if the matter fields satisf y the dominant energy condition. By using these results, we analytically sh ow that once gravitational collapse occurs from any initial data on a null hypersurface, the system of field equations breaks down inevitably in the d omain of outer communications or the boundary, i.e. the black hole event ho rizon provided that a future null infinity I+ exists, or the asymptotic str ucture at I+ is broken and the universe will recollapse. In order to clarif y which history does the universe trace, we perform a numerical simulation. Then, the dilaton field diverges faster than the logarithmic function almo st uniformly and the asymptotic structure would be broken. This implies tha t the cosmic no hair conjecture is violated in the generalized theory of gr avity.